If by principal $\mathbf{G}_a$-bundle you mean a $\mathbf{G}_a$-torsor in the fppf topology, then this can proved as follows. Since such objects are classified by elements in the fppf cohomology of your scheme with coefficients in $\mathbf{G}_a$, the result follows from the following proposition.

**Proposition:** Let $S$ be an affine scheme. Then
$$H^1_{\text{fppf}}(S, \mathbf{G}_a) = 0.$$

**Proof:** Since $\mathbf{G}_a$ is smooth (over $S$), by a theorem of Grothendieck from Brauer III we  have $$H^1_{\text{fppf}}(S, \mathbf{G}_a) = H^1_{\text{ét}}(S, \mathbf{G}_a).$$ 
On the other hand, by Theorem 2.1 [here](http://virtualmath1.stanford.edu/~conrad/Weil2seminar/Notes/L4.pdf) on the comparison between Zariski and étale cohomology,  
$$H^1_{\text{ét}}(S, \mathbf{G}_a) = H^1(S, \mathcal{O}_S).$$ The group on the right vanishes since $S$ is affine, proving the proposition.